I do a lot of venison on the grill. Just my opinion, but I would suggest leaving the back strap whole and slice after it's cooked.
I marinade the whole back strap in Stubbs Beef Marinade in the fridge, overnight.
I have a bit different approach from some others, in that I don't use high heat until the end. I use Weber gas grills and I start out with settings all set on the lowest setting. I play it by eye, so I don't have an exact temp. I put the whole strap on for about 10 minutes a side. Again, on low, away from direct heat. This allows for more uniform cooking. With a whole loin, it's easy to get a well done outer layer and a nearly raw inner core. When I do it this way, the very outside is medium and inner core is a uniform, cooked pink medium rare.
Use tongs for flipping. Do not pierce with a fork.
Then I crank the heat up to high and cook for about 2-3 minutes a side. This gives a crispy outside and uniform, prime rib like juicy medium rare in the middle.
I remove the back strap, plate it and cover with foil for at least 5 minutes. Do not cut for at least 5 minutes or all the juices will flow out.
Then slice into about 1" slices and hit it with a sea salt grinder, to taste. Pepper isn't needed or even suggested, in my opinion.
Hope this helps.